4. Applications of Derivatives

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ (log₂ x - log₃ x)

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π/2⁻ (π - 2x) tan x  

5–7. For each function ƒ and interval [a, b], a graph of ƒ is given along with the secant line that passes though the graph of ƒ at x = a and x = b.

a. Use the graph to make a conjecture about the value(s) of c satisfying the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) .

b. Verify your answer to part (a) by solving the equation (ƒ(b) - ƒ(a)) / (b-a) = ƒ' (c) for c.

ƒ(x) = x² / 4 + 1 ; [ -2, 4] <IMAGE>

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0⁺ (x - 3 √x) / (x - √x)

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_t→0 (1 - cos 6t) / 2t

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_Θ→0 (3 sin² 2Θ) / Θ²

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_Θ→0 2Θ cot 3Θ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_y→0⁺ (ln¹⁰ y) / √y

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→1 (x⁴ - x³ - 3x² + 5x -2) / x³ + x² - 5x + 3

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→∞ x³ (1/x - sin 1/x)

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0⁺ x²ˣ

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_Θ→π/2⁻ (tan Θ)ᶜᵒˢ ᶿ

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0 (x + cos x)¹/ˣ

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_x→0 csc x sin⁻¹ x